Model theory
Read as: c sub zero
Means: The expression reads: c sub zero. It names the displayed model-theoretic object in the surrounding argument. Its 1 exact source occurrence was reviewed in the compactness argument for an infinite model.
Read as: relation I sub n plus one open parenthesis sequence a comma sequence b close parenthesis
Means: The expression reads: relation I sub n plus one open parenthesis sequence a comma sequence b close parenthesis. It belongs to the finite back-and-forth analysis that connects sequence relations with formulas of bounded quantifier rank. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: i equals one
Means: The expression reads: i equals one. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in dense linear orders without endpoints.
Read as: n equals zero
Means: The expression reads: n equals zero. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 2 exact source occurrences were reviewed in partial isomorphisms and bounded quantifier rank.
Read as: structure Q has domain the rational numbers with the strict order relation
Means: The expression reads: structure Q has domain the rational numbers with the strict order relation. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in dense linear orders without endpoints.
Read as: s
Means: The expression reads: s. It names the displayed model-theoretic object in the surrounding argument. Its 4 exact source occurrences were reviewed in elementary equivalence and isomorphism and partial isomorphisms and bounded quantifier rank.
Read as: a sub n
Means: The expression reads: a sub n. It names the displayed model-theoretic object in the surrounding argument. Its 4 exact source occurrences were reviewed in partial isomorphisms and bounded quantifier rank.
Read as: left angle bracket a sub one comma and so on comma a sub n right angle bracket is in the interpretation of R in structure M prime
Means: The expression reads: left angle bracket a sub one comma and so on comma a sub n right angle bracket is in the interpretation of R in structure M prime. It identifies an interpretation or term value in a named structure and assignment. Its 1 exact source occurrence was reviewed in substructures.
Read as: the quantifier rank of formula A is less than or equal to n
Means: The expression reads: the quantifier rank of formula A is less than or equal to n. It belongs to the finite back-and-forth analysis that connects sequence relations with formulas of bounded quantifier rank. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: the domain of p
Means: The expression reads: the domain of p. It denotes a structure domain, a function domain, or the finite sequences drawn from that domain. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: structure M is back-and-forth equivalent at level n to structure N
Means: The expression reads: structure M is back-and-forth equivalent at level n to structure N. It belongs to the finite back-and-forth analysis that connects sequence relations with formulas of bounded quantifier rank. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: x sub n
Means: The expression reads: x sub n. It names the displayed model-theoretic object in the surrounding argument. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: structure R is partially isomorphic to structure Q
Means: The expression reads: structure R is partially isomorphic to structure Q. It compares structures by isomorphism, partial isomorphism, or agreement on first-order sentences, exactly as the notation indicates. Its 1 exact source occurrence was reviewed in dense linear orders without endpoints.
Read as: the domain of structure M
Means: The expression reads: the domain of structure M. It denotes a structure domain, a function domain, or the finite sequences drawn from that domain. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: the rational numbers
Means: The expression reads: the rational numbers. It names the displayed model-theoretic object in the surrounding argument. Its 1 exact source occurrence was reviewed in dense linear orders without endpoints.
Read as: structure S is isomorphic to structure Q
Means: The expression reads: structure S is isomorphic to structure Q. It compares structures by isomorphism, partial isomorphism, or agreement on first-order sentences, exactly as the notation indicates. Its 1 exact source occurrence was reviewed in dense linear orders without endpoints.
Read as: the domain of structure M equals the domain of structure M prime
Means: The expression reads: the domain of structure M equals the domain of structure M prime. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in reducts and expansions.
Read as: formula T superscript a sub n
Means: The expression reads: formula T superscript a sub n. It names the displayed model-theoretic object in the surrounding argument. Its 3 exact source occurrences were reviewed in partial isomorphisms and bounded quantifier rank.
Read as: s sub two open parenthesis x sub i close parenthesis equals b sub i
Means: The expression reads: s sub two open parenthesis x sub i close parenthesis equals b sub i. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: the value of x in structure M at assignment s equals s open parenthesis x close parenthesis
Means: The expression reads: the value of x in structure M at assignment s equals s open parenthesis x close parenthesis. It identifies an interpretation or term value in a named structure and assignment. Its 1 exact source occurrence was reviewed in elementary equivalence and isomorphism.
Read as: a sub r
Means: The expression reads: a sub r. It names the displayed model-theoretic object in the surrounding argument. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: sequence a is a finite sequence over the domain of structure M
Means: The expression reads: sequence a is a finite sequence over the domain of structure M. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: c is in language L
Means: The expression reads: c is in language L. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 2 exact source occurrences were reviewed in reducts and expansions and substructures.
Read as: b is in the domain of structure M sub two
Means: The expression reads: b is in the domain of structure M sub two. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 4 exact source occurrences were reviewed in dense linear orders without endpoints.
Read as: structure M is isomorphic to structure M prime
Means: The expression reads: structure M is isomorphic to structure M prime. It compares structures by isomorphism, partial isomorphism, or agreement on first-order sentences, exactly as the notation indicates. Its 1 exact source occurrence was reviewed in elementary equivalence and isomorphism.
Read as: a sub one
Means: The expression reads: a sub one. It names the displayed model-theoretic object in the surrounding argument. Its 6 exact source occurrences were reviewed in substructures and partial isomorphisms and bounded quantifier rank.
Read as: p
Means: The expression reads: p. It names the displayed model-theoretic object in the surrounding argument. Its 10 exact source occurrences were reviewed in partial isomorphisms and bounded quantifier rank and dense linear orders without endpoints.
Read as: relation I sub zero open parenthesis sequence a comma sequence b close parenthesis
Means: The expression reads: relation I sub zero open parenthesis sequence a comma sequence b close parenthesis. It belongs to the finite back-and-forth analysis that connects sequence relations with formulas of bounded quantifier rank. Its 2 exact source occurrences were reviewed in partial isomorphisms and bounded quantifier rank.
Read as: h open parenthesis the value of t in structure M at assignment s close parenthesis equals the value of t in structure M prime at assignment h composed with s
Means: The expression reads: h open parenthesis the value of t in structure M at assignment s close parenthesis equals the value of t in structure M prime at assignment h composed with s. It identifies an interpretation or term value in a named structure and assignment. Its 1 exact source occurrence was reviewed in elementary equivalence and isomorphism.
Read as: N is a subset of the domain of structure M
Means: The expression reads: N is a subset of the domain of structure M. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in substructures.
Read as: relation I sub n open parenthesis sequence a comma sequence b close parenthesis
Means: The expression reads: relation I sub n open parenthesis sequence a comma sequence b close parenthesis. It belongs to the finite back-and-forth analysis that connects sequence relations with formulas of bounded quantifier rank. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: structure M sub two
Means: The expression reads: structure M sub two. It names the displayed model-theoretic object in the surrounding argument. Its 2 exact source occurrences were reviewed in dense linear orders without endpoints.
Read as: s sub one open parenthesis x sub i close parenthesis equals a sub i
Means: The expression reads: s sub one open parenthesis x sub i close parenthesis equals a sub i. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: n
Means: The expression reads: n. It names the displayed model-theoretic object in the surrounding argument. Its 24 exact source occurrences were reviewed in reducts and expansions, substructures, the compactness argument for an infinite model, elementary equivalence and isomorphism, partial isomorphisms and bounded quantifier rank and dense linear orders without endpoints.
Read as: formula A is in the theory of structure M
Means: The expression reads: formula A is in the theory of structure M. It identifies a theory, a set defined by a condition, or membership in such a set. Its 4 exact source occurrences were reviewed in the theory of a structure.
Read as: for every x, for every y, open parenthesis x is less than y or x equals y or y is less than x close parenthesis
Means: The expression reads: for every x, for every y, open parenthesis x is less than y or x equals y or y is less than x close parenthesis. It is a complete first-order condition; quantifier scope and connective order are spoken explicitly. Its 1 exact source occurrence was reviewed in dense linear orders without endpoints.
Read as: the theory of structure R
Means: The expression reads: the theory of structure R. It identifies a theory, a set defined by a condition, or membership in such a set. Its 1 exact source occurrence was reviewed in the theory of a structure.
Read as: n plus one
Means: The expression reads: n plus one. It names the displayed model-theoretic object in the surrounding argument. Its 5 exact source occurrences were reviewed in partial isomorphisms and bounded quantifier rank.
Read as: h open parenthesis the value of t sub i in structure M at assignment s close parenthesis equals the value of t sub i in structure M prime at assignment h composed with s
Means: The expression reads: h open parenthesis the value of t sub i in structure M at assignment s close parenthesis equals the value of t sub i in structure M prime at assignment h composed with s. It identifies an interpretation or term value in a named structure and assignment. Its 1 exact source occurrence was reviewed in elementary equivalence and isomorphism.
Read as: a is in the domain of structure M sub one
Means: The expression reads: a is in the domain of structure M sub one. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in dense linear orders without endpoints.
Read as: structure M
Means: The expression reads: structure M. It names the displayed model-theoretic object in the surrounding argument. Its 10 exact source occurrences were reviewed in reducts and expansions, substructures, the compactness argument for an infinite model and the theory of a structure.
Read as: the value of c in structure M at assignment s equals the interpretation of c in structure M
Means: The expression reads: the value of c in structure M at assignment s equals the interpretation of c in structure M. It identifies an interpretation or term value in a named structure and assignment. Its 1 exact source occurrence was reviewed in elementary equivalence and isomorphism.
Read as: a sub n is less than sub one a
Means: The expression reads: a sub n is less than sub one a. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in dense linear orders without endpoints.
Read as: f
Means: The expression reads: f. It names the displayed model-theoretic object in the surrounding argument. Its 2 exact source occurrences were reviewed in elementary equivalence and isomorphism and partial isomorphisms and bounded quantifier rank.
Read as: m is greater than zero
Means: The expression reads: m is greater than zero. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in the compactness argument for an infinite model.
Read as: p open parenthesis the interpretation of f in structure M open parenthesis a sub one comma and so on comma a sub n close parenthesis close parenthesis equals the interpretation of f in structure N open parenthesis p open parenthesis a sub one close parenthesis comma and so on comma p open parenthesis a sub n close parenthesis close parenthesis
Means: The expression reads: p open parenthesis the interpretation of f in structure M open parenthesis a sub one comma and so on comma a sub n close parenthesis close parenthesis equals the interpretation of f in structure N open parenthesis p open parenthesis a sub one close parenthesis comma and so on comma p open parenthesis a sub n close parenthesis close parenthesis. It identifies an interpretation or term value in a named structure and assignment. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: the theory of structure M equals the set of formula A such that structure M satisfies formula A
Means: The expression reads: the theory of structure M equals the set of formula A such that structure M satisfies formula A. It identifies a theory, a set defined by a condition, or membership in such a set. Its 1 exact source occurrence was reviewed in the theory of a structure.
Read as: R is a subset of the domain of structure M superscript n
Means: The expression reads: R is a subset of the domain of structure M superscript n. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in reducts and expansions.
Read as: c sub one
Means: The expression reads: c sub one. It names the displayed model-theoretic object in the surrounding argument. Its 1 exact source occurrence was reviewed in the compactness argument for an infinite model.
Read as: structure M satisfies formula A at assignment s
Means: The expression reads: structure M satisfies formula A at assignment s. It asserts truth or failure of truth in the named structure, with the displayed assignment when one is present. Its 1 exact source occurrence was reviewed in elementary equivalence and isomorphism.
Read as: for every x, not x is less than x
Means: The expression reads: for every x, not x is less than x. It is a complete first-order condition; quantifier scope and connective order are spoken explicitly. Its 1 exact source occurrence was reviewed in dense linear orders without endpoints.
Read as: a is in the domain of structure M
Means: The expression reads: a is in the domain of structure M. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: c
Means: The expression reads: c. It names the displayed model-theoretic object in the surrounding argument. Its 2 exact source occurrences were reviewed in elementary equivalence and isomorphism and partial isomorphisms and bounded quantifier rank.
Read as: q belongs to the family I of partial isomorphisms
Means: The expression reads: q belongs to the family I of partial isomorphisms. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in dense linear orders without endpoints.
Read as: i equals one
Means: The expression reads: i equals one. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: h composed with s
Means: The expression reads: h composed with s. It names the displayed model-theoretic object in the surrounding argument. Its 2 exact source occurrences were reviewed in elementary equivalence and isomorphism.
Read as: the domain of structure M
Means: The expression reads: the domain of structure M. It denotes a structure domain, a function domain, or the finite sequences drawn from that domain. Its 3 exact source occurrences were reviewed in substructures, the compactness argument for an infinite model and partial isomorphisms and bounded quantifier rank.
Read as: structure M sub one is isomorphic to structure M sub two
Means: The expression reads: structure M sub one is isomorphic to structure M sub two. It compares structures by isomorphism, partial isomorphism, or agreement on first-order sentences, exactly as the notation indicates. Its 1 exact source occurrence was reviewed in dense linear orders without endpoints.
Read as: structure M is elementarily equivalent to structure M prime
Means: The expression reads: structure M is elementarily equivalent to structure M prime. It compares structures by isomorphism, partial isomorphism, or agreement on first-order sentences, exactly as the notation indicates. Its 1 exact source occurrence was reviewed in elementary equivalence and isomorphism.
Read as: x is in the domain of structure M
Means: The expression reads: x is in the domain of structure M. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in elementary equivalence and isomorphism.
Read as: b
Means: The expression reads: b. It names the displayed model-theoretic object in the surrounding argument. Its 2 exact source occurrences were reviewed in partial isomorphisms and bounded quantifier rank and dense linear orders without endpoints.
Read as: p sub n is a subset of p sub n plus one
Means: The expression reads: p sub n is a subset of p sub n plus one. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 2 exact source occurrences were reviewed in partial isomorphisms and bounded quantifier rank.
Read as: language L prime
Means: The expression reads: language L prime. It names the displayed model-theoretic object in the surrounding argument. Its 4 exact source occurrences were reviewed in reducts and expansions.
Read as: structure M
Means: The expression reads: structure M. It names the displayed model-theoretic object in the surrounding argument. Its 39 exact source occurrences were reviewed in reducts and expansions, substructures, elementary equivalence and isomorphism, the theory of a structure, partial isomorphisms and bounded quantifier rank and dense linear orders without endpoints.
Read as: finite sequence a followed by element a
Means: The expression reads: finite sequence a followed by element a. It names the displayed model-theoretic object in the surrounding argument. Its 5 exact source occurrences were reviewed in partial isomorphisms and bounded quantifier rank.
Read as: n plus one equals two times r
Means: The expression reads: n plus one equals two times r. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: first row: the value of t in structure M at assignment s equals the interpretation of f in M applied to the values of t sub one through t sub n in M at s; second row: the frozen source prints the value of t in M prime at h composed with s as using the interpretation of f in M. That structure is wrong; the disclosed reader correction uses the interpretation of f in M prime
Means: The expression reads: first row: the value of t in structure M at assignment s equals the interpretation of f in M applied to the values of t sub one through t sub n in M at s; second row: the frozen source prints the value of t in M prime at h composed with s as using the interpretation of f in M. That structure is wrong; the disclosed reader correction uses the interpretation of f in M prime. It identifies an interpretation or term value in a named structure and assignment. Its 1 exact source occurrence was reviewed in elementary equivalence and isomorphism. The frozen second row incorrectly uses the interpretation of f in M. The separately labelled reader MathML and speech use M prime while retaining the source form.
Read as: h open parenthesis x close parenthesis equals y
Means: The expression reads: h open parenthesis x close parenthesis equals y. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in elementary equivalence and isomorphism.
Read as: structure M satisfies formula A at assignment s sub one
Means: The expression reads: structure M satisfies formula A at assignment s sub one. It asserts truth or failure of truth in the named structure, with the displayed assignment when one is present. Its 2 exact source occurrences were reviewed in partial isomorphisms and bounded quantifier rank.
Read as: b is in the domain of structure N
Means: The expression reads: b is in the domain of structure N. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 5 exact source occurrences were reviewed in partial isomorphisms and bounded quantifier rank.
Read as: for every x, for every y, open parenthesis x is less than y implies there exists z, open parenthesis x is less than z and z is less than y close parenthesis close parenthesis
Means: The expression reads: for every x, for every y, open parenthesis x is less than y implies there exists z, open parenthesis x is less than z and z is less than y close parenthesis close parenthesis. It is a complete first-order condition; quantifier scope and connective order are spoken explicitly. Its 1 exact source occurrence was reviewed in dense linear orders without endpoints.
Read as: structure M is elementarily equivalent to structure M prime
Means: The expression reads: structure M is elementarily equivalent to structure M prime. It compares structures by isomorphism, partial isomorphism, or agreement on first-order sentences, exactly as the notation indicates. Its 1 exact source occurrence was reviewed in elementary equivalence and isomorphism.
Read as: X
Means: The expression reads: X. It names the displayed model-theoretic object in the surrounding argument. Its 2 exact source occurrences were reviewed in elementary equivalence and isomorphism.
Read as: R is in language L
Means: The expression reads: R is in language L. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in substructures.
Read as: for every x, there exists y, x is less than y
Means: The expression reads: for every x, there exists y, x is less than y. It is a complete first-order condition; quantifier scope and connective order are spoken explicitly. Its 1 exact source occurrence was reviewed in dense linear orders without endpoints.
Read as: p sub zero is contained in p sub one which is contained in p sub two and the inclusion chain continues in the same way
Means: The expression reads: p sub zero is contained in p sub one which is contained in p sub two and the inclusion chain continues in the same way. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: p sub n plus one
Means: The expression reads: p sub n plus one. It names the displayed model-theoretic object in the surrounding argument. Its 2 exact source occurrences were reviewed in partial isomorphisms and bounded quantifier rank.
Read as: structure R has domain the real numbers with the strict order relation
Means: The expression reads: structure R has domain the real numbers with the strict order relation. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in the theory of a structure.
Read as: c sub i is not equal to c sub j
Means: The expression reads: c sub i is not equal to c sub j. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in the compactness argument for an infinite model.
Read as: structure M is back-and-forth equivalent at level n to structure N
Means: The expression reads: structure M is back-and-forth equivalent at level n to structure N. It belongs to the finite back-and-forth analysis that connects sequence relations with formulas of bounded quantifier rank. Its 2 exact source occurrences were reviewed in partial isomorphisms and bounded quantifier rank.
Read as: structure M sub one is partially isomorphic to structure M sub two
Means: The expression reads: structure M sub one is partially isomorphic to structure M sub two. It compares structures by isomorphism, partial isomorphism, or agreement on first-order sentences, exactly as the notation indicates. Its 2 exact source occurrences were reviewed in dense linear orders without endpoints.
Read as: b sub n
Means: The expression reads: b sub n. It names the displayed model-theoretic object in the surrounding argument. Its 2 exact source occurrences were reviewed in partial isomorphisms and bounded quantifier rank.
Read as: structure N satisfies formula A at assignment sequence b
Means: The expression reads: structure N satisfies formula A at assignment sequence b. It asserts truth or failure of truth in the named structure, with the displayed assignment when one is present. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: relation I sub n is contained in the product of the finite sequences over the domains of structures M and N
Means: The expression reads: relation I sub n is contained in the product of the finite sequences over the domains of structures M and N. It belongs to the finite back-and-forth analysis that connects sequence relations with formulas of bounded quantifier rank. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: a sub n is in the domain of structure M
Means: The expression reads: a sub n is in the domain of structure M. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 2 exact source occurrences were reviewed in substructures.
Read as: Gamma
Means: The expression reads: Gamma. It names the displayed model-theoretic object in the surrounding argument. Its 4 exact source occurrences were reviewed in the compactness argument for an infinite model.
Read as: p open parenthesis the interpretation of c in structure M close parenthesis
Means: The expression reads: p open parenthesis the interpretation of c in structure M close parenthesis. It identifies an interpretation or term value in a named structure and assignment. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: the value of c in structure M prime at assignment h composed with s equals the interpretation of c in structure M prime
Means: The expression reads: the value of c in structure M prime at assignment h composed with s equals the interpretation of c in structure M prime. It identifies an interpretation or term value in a named structure and assignment. Its 1 exact source occurrence was reviewed in elementary equivalence and isomorphism.
Read as: s open parenthesis x sub i close parenthesis equals a sub i
Means: The expression reads: s open parenthesis x sub i close parenthesis equals a sub i. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: n is in the natural numbers
Means: The expression reads: n is in the natural numbers. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: structure N
Means: The expression reads: structure N. It names the displayed model-theoretic object in the surrounding argument. Its 16 exact source occurrences were reviewed in substructures and partial isomorphisms and bounded quantifier rank.
Read as: P
Means: The expression reads: P. It names the displayed model-theoretic object in the surrounding argument. Its 3 exact source occurrences were reviewed in reducts and expansions, elementary equivalence and isomorphism and partial isomorphisms and bounded quantifier rank.
Read as: formula B open parenthesis x comma bold y close parenthesis
Means: The expression reads: formula B open parenthesis x comma bold y close parenthesis. It names the displayed model-theoretic object in the surrounding argument. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: structure S
Means: The expression reads: structure S. It names the displayed model-theoretic object in the surrounding argument. Its 5 exact source occurrences were reviewed in the theory of a structure and dense linear orders without endpoints.
Read as: f is in language L
Means: The expression reads: f is in language L. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 2 exact source occurrences were reviewed in reducts and expansions and substructures.
Read as: structure N satisfies formula A
Means: The expression reads: structure N satisfies formula A. It asserts truth or failure of truth in the named structure, with the displayed assignment when one is present. Its 2 exact source occurrences were reviewed in the theory of a structure.
Read as: a is less than sub one a sub one
Means: The expression reads: a is less than sub one a sub one. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in dense linear orders without endpoints.
Read as: the tuple a sub one comma and so on comma a sub n is in the interpretation of P in structure M iff the tuple h open parenthesis a sub one close parenthesis comma and so on comma h open parenthesis a sub n close parenthesis is in the interpretation of P in structure M prime semicolon
Means: The expression reads: the tuple a sub one comma and so on comma a sub n is in the interpretation of P in structure M iff the tuple h open parenthesis a sub one close parenthesis comma and so on comma h open parenthesis a sub n close parenthesis is in the interpretation of P in structure M prime semicolon. It identifies an interpretation or term value in a named structure and assignment. Its 1 exact source occurrence was reviewed in elementary equivalence and isomorphism.
Read as: the order relation less than sub two equals the interpretation of the order relation less than in structure M sub two
Means: The expression reads: the order relation less than sub two equals the interpretation of the order relation less than in structure M sub two. It identifies an interpretation or term value in a named structure and assignment. Its 1 exact source occurrence was reviewed in dense linear orders without endpoints.
Read as: p sub zero is in I
Means: The expression reads: p sub zero is in I. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: there exists x, formula T superscript a sub n open parenthesis x comma bold y close parenthesis
Means: The expression reads: there exists x, formula T superscript a sub n open parenthesis x comma bold y close parenthesis. It is a complete first-order condition; quantifier scope and connective order are spoken explicitly. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: a sub i
Means: The expression reads: a sub i. It names the displayed model-theoretic object in the surrounding argument. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: q equals the union of p and the singleton set containing the ordered pair a and b
Means: The expression reads: q equals the union of p and the singleton set containing the ordered pair a and b. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in dense linear orders without endpoints.
Read as: structure R equals the ordered structure with domain the real numbers and strict order relation less than
Means: The expression reads: structure R equals the ordered structure with domain the real numbers and strict order relation less than. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in dense linear orders without endpoints.
Read as: the real numbers
Means: The expression reads: the real numbers. It names the displayed model-theoretic object in the surrounding argument. Its 1 exact source occurrence was reviewed in the theory of a structure.
Read as: left angle bracket a sub one comma and so on comma a sub n right angle bracket is in the interpretation of R in structure M
Means: The expression reads: left angle bracket a sub one comma and so on comma a sub n right angle bracket is in the interpretation of R in structure M. It identifies an interpretation or term value in a named structure and assignment. Its 1 exact source occurrence was reviewed in substructures.
Read as: b sub one
Means: The expression reads: b sub one. It names the displayed model-theoretic object in the surrounding argument. Its 2 exact source occurrences were reviewed in partial isomorphisms and bounded quantifier rank.
Read as: p equals the union over all n greater than or equal to zero of p sub n
Means: The expression reads: p equals the union over all n greater than or equal to zero of p sub n. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: the family I of partial isomorphisms
Means: The expression reads: the family I of partial isomorphisms. It names the displayed model-theoretic object in the surrounding argument. Its 5 exact source occurrences were reviewed in dense linear orders without endpoints.
Read as: p open parenthesis the interpretation of c in structure M close parenthesis equals the interpretation of c in structure N
Means: The expression reads: p open parenthesis the interpretation of c in structure M close parenthesis equals the interpretation of c in structure N. It identifies an interpretation or term value in a named structure and assignment. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: p belongs to the family I of partial isomorphisms
Means: The expression reads: p belongs to the family I of partial isomorphisms. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in dense linear orders without endpoints.
Read as: left angle bracket p open parenthesis a sub one close parenthesis comma and so on comma p open parenthesis a sub n close parenthesis right angle bracket is in the interpretation of P in structure N
Means: The expression reads: left angle bracket p open parenthesis a sub one close parenthesis comma and so on comma p open parenthesis a sub n close parenthesis right angle bracket is in the interpretation of P in structure N. It identifies an interpretation or term value in a named structure and assignment. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: sequence a
Means: The expression reads: sequence a. It names the displayed model-theoretic object in the surrounding argument. Its 7 exact source occurrences were reviewed in partial isomorphisms and bounded quantifier rank.
Read as: the empty sequence
Means: The expression reads: the empty sequence. It names the displayed model-theoretic object in the surrounding argument. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: the theory of structure N is a subset of the theory of structure M
Means: The expression reads: the theory of structure N is a subset of the theory of structure M. It identifies a theory, a set defined by a condition, or membership in such a set. Its 1 exact source occurrence was reviewed in the theory of a structure.
Read as: formula A
Means: The expression reads: formula A. It names the displayed model-theoretic object in the surrounding argument. Its 15 exact source occurrences were reviewed in reducts and expansions, the compactness argument for an infinite model, elementary equivalence and isomorphism, the theory of a structure and partial isomorphisms and bounded quantifier rank.
Read as: structure M sub one
Means: The expression reads: structure M sub one. It names the displayed model-theoretic object in the surrounding argument. Its 1 exact source occurrence was reviewed in dense linear orders without endpoints.
Read as: p is a subset of q
Means: The expression reads: p is a subset of q. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 2 exact source occurrences were reviewed in partial isomorphisms and bounded quantifier rank.
Read as: structure M prime satisfies formula A
Means: The expression reads: structure M prime satisfies formula A. It asserts truth or failure of truth in the named structure, with the displayed assignment when one is present. Its 2 exact source occurrences were reviewed in elementary equivalence and isomorphism.
Read as: a is in the domain of structure M
Means: The expression reads: a is in the domain of structure M. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 4 exact source occurrences were reviewed in partial isomorphisms and bounded quantifier rank.
Read as: the value of x in structure M prime at assignment h composed with s equals h open parenthesis s open parenthesis x close parenthesis close parenthesis
Means: The expression reads: the value of x in structure M prime at assignment h composed with s equals h open parenthesis s open parenthesis x close parenthesis close parenthesis. It identifies an interpretation or term value in a named structure and assignment. Its 1 exact source occurrence was reviewed in elementary equivalence and isomorphism.
Read as: the empty set
Means: The expression reads: the empty set. It names the displayed model-theoretic object in the surrounding argument. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: the interpretation of f in structure M open parenthesis a sub one comma and so on comma a sub n close parenthesis equals the interpretation of f in structure M prime open parenthesis a sub one comma and so on comma a sub n close parenthesis
Means: The expression reads: the interpretation of f in structure M open parenthesis a sub one comma and so on comma a sub n close parenthesis equals the interpretation of f in structure M prime open parenthesis a sub one comma and so on comma a sub n close parenthesis. It identifies an interpretation or term value in a named structure and assignment. Its 1 exact source occurrence was reviewed in substructures.
Read as: q
Means: The expression reads: q. It names the displayed model-theoretic object in the surrounding argument. Its 2 exact source occurrences were reviewed in partial isomorphisms and bounded quantifier rank.
Read as: sequence b
Means: The expression reads: sequence b. It names the displayed model-theoretic object in the surrounding argument. Its 5 exact source occurrences were reviewed in partial isomorphisms and bounded quantifier rank.
Read as: the domain of structure N
Means: The expression reads: the domain of structure N. It denotes a structure domain, a function domain, or the finite sequences drawn from that domain. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: structure M is elementarily equivalent through quantifier rank n to structure N
Means: The expression reads: structure M is elementarily equivalent through quantifier rank n to structure N. It compares structures by isomorphism, partial isomorphism, or agreement on first-order sentences, exactly as the notation indicates. Its 2 exact source occurrences were reviewed in partial isomorphisms and bounded quantifier rank.
Read as: p open parenthesis a sub i close parenthesis equals b sub i
Means: The expression reads: p open parenthesis a sub i close parenthesis equals b sub i. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in dense linear orders without endpoints.
Read as: relation I sub n holds of the sequence formed by appending a to sequence a and the sequence formed by appending b to sequence b
Means: The expression reads: relation I sub n holds of the sequence formed by appending a to sequence a and the sequence formed by appending b to sequence b. It belongs to the finite back-and-forth analysis that connects sequence relations with formulas of bounded quantifier rank. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: structure M prime
Means: The expression reads: structure M prime. It names the displayed model-theoretic object in the surrounding argument. Its 11 exact source occurrences were reviewed in reducts and expansions, substructures and elementary equivalence and isomorphism.
Read as: the frozen first row prints h applied to the value of t in M at s equals h applied to the interpretation of f in M at the component term values, but leaves the outer h unclosed. The disclosed reader correction closes it; the following rows use preservation of f, the induction hypothesis for each component term, and conclude with the value of t in M prime at h composed with s
Means: The expression reads: the frozen first row prints h applied to the value of t in M at s equals h applied to the interpretation of f in M at the component term values, but leaves the outer h unclosed. The disclosed reader correction closes it; the following rows use preservation of f, the induction hypothesis for each component term, and conclude with the value of t in M prime at h composed with s. It identifies an interpretation or term value in a named structure and assignment. Its 1 exact source occurrence was reviewed in elementary equivalence and isomorphism. The frozen first equality omits the closing parenthesis for the outer application of h. The separately labelled reader MathML supplies it while retaining the source form.
Read as: structure M satisfies formula A iff structure M prime satisfies formula A
Means: The expression reads: structure M satisfies formula A iff structure M prime satisfies formula A. It asserts truth or failure of truth in the named structure, with the displayed assignment when one is present. Its 1 exact source occurrence was reviewed in reducts and expansions.
Read as: t is syntactically identical to x
Means: The expression reads: t is syntactically identical to x. It states which constant, variable, or compound term the term t is syntactically identical to. Its 1 exact source occurrence was reviewed in elementary equivalence and isomorphism.
Read as: sequence a comma sequence b comma sequence c comma and so on
Means: The expression reads: sequence a comma sequence b comma sequence c comma and so on. It names the displayed model-theoretic object in the surrounding argument. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: structure M is a substructure of structure M prime
Means: The expression reads: structure M is a substructure of structure M prime. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in substructures.
Read as: s sub one open parenthesis x sub i close parenthesis equals a sub i
Means: The expression reads: s sub one open parenthesis x sub i close parenthesis equals a sub i. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: b is less than sub two b sub one
Means: The expression reads: b is less than sub two b sub one. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in dense linear orders without endpoints.
Read as: equals the value of t in structure M prime at assignment h composed with s
Means: The expression reads: equals the value of t in structure M prime at assignment h composed with s. It identifies an interpretation or term value in a named structure and assignment. Its 1 exact source occurrence was reviewed in elementary equivalence and isomorphism.
Read as: structure N satisfies formula A at assignment s sub two
Means: The expression reads: structure N satisfies formula A at assignment s sub two. It asserts truth or failure of truth in the named structure, with the displayed assignment when one is present. Its 2 exact source occurrences were reviewed in partial isomorphisms and bounded quantifier rank.
Read as: a sub one is less than sub one a sub two and the increasing chain continues in order through a sub n
Means: The expression reads: a sub one is less than sub one a sub two and the increasing chain continues in order through a sub n. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in dense linear orders without endpoints.
Read as: b sub n is less than sub two b
Means: The expression reads: b sub n is less than sub two b. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in dense linear orders without endpoints.
Read as: t is syntactically identical to c
Means: The expression reads: t is syntactically identical to c. It states which constant, variable, or compound term the term t is syntactically identical to. Its 1 exact source occurrence was reviewed in elementary equivalence and isomorphism.
Read as: b sub i
Means: The expression reads: b sub i. It names the displayed model-theoretic object in the surrounding argument. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: the domain of structure M is a subset of the domain of structure M prime
Means: The expression reads: the domain of structure M is a subset of the domain of structure M prime. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 2 exact source occurrences were reviewed in substructures.
Read as: structure M is elementarily equivalent to structure N
Means: The expression reads: structure M is elementarily equivalent to structure N. It compares structures by isomorphism, partial isomorphism, or agreement on first-order sentences, exactly as the notation indicates. Its 5 exact source occurrences were reviewed in the theory of a structure and partial isomorphisms and bounded quantifier rank.
Read as: h open parenthesis the value of x in structure M at assignment s close parenthesis equals h open parenthesis s open parenthesis x close parenthesis close parenthesis equals the value of x in structure M prime at assignment h composed with s
Means: The expression reads: h open parenthesis the value of x in structure M at assignment s close parenthesis equals h open parenthesis s open parenthesis x close parenthesis close parenthesis equals the value of x in structure M prime at assignment h composed with s. It identifies an interpretation or term value in a named structure and assignment. Its 1 exact source occurrence was reviewed in elementary equivalence and isomorphism.
Read as: structure R
Means: The expression reads: structure R. It names the displayed model-theoretic object in the surrounding argument. Its 2 exact source occurrences were reviewed in the theory of a structure.
Read as: b is in the domain of structure N
Means: The expression reads: b is in the domain of structure N. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: the negation of formula A does not belong to the theory of structure M
Means: The expression reads: the negation of formula A does not belong to the theory of structure M. It identifies a theory, a set defined by a condition, or membership in such a set. Its 1 exact source occurrence was reviewed in the theory of a structure.
Read as: I
Means: The expression reads: I. It names the displayed model-theoretic object in the surrounding argument. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: h
Means: The expression reads: h. It names the displayed model-theoretic object in the surrounding argument. Its 6 exact source occurrences were reviewed in elementary equivalence and isomorphism.
Read as: language L is a subset of language L prime
Means: The expression reads: language L is a subset of language L prime. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in reducts and expansions.
Read as: b sub r
Means: The expression reads: b sub r. It names the displayed model-theoretic object in the surrounding argument. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: h open parenthesis x close parenthesis equals h open parenthesis y close parenthesis
Means: The expression reads: h open parenthesis x close parenthesis equals h open parenthesis y close parenthesis. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in elementary equivalence and isomorphism.
Read as: structure N does not satisfy the negation of formula A
Means: The expression reads: structure N does not satisfy the negation of formula A. It asserts truth or failure of truth in the named structure, with the displayed assignment when one is present. Its 1 exact source occurrence was reviewed in the theory of a structure.
Read as: language L is a subset of language L prime
Means: The expression reads: language L is a subset of language L prime. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in reducts and expansions.
Read as: x equals y
Means: The expression reads: x equals y. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in elementary equivalence and isomorphism.
Read as: relation I sub n open parenthesis the empty sequence comma the empty sequence close parenthesis
Means: The expression reads: relation I sub n open parenthesis the empty sequence comma the empty sequence close parenthesis. It belongs to the finite back-and-forth analysis that connects sequence relations with formulas of bounded quantifier rank. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: language L
Means: The expression reads: language L. It names the displayed model-theoretic object in the surrounding argument. Its 14 exact source occurrences were reviewed in reducts and expansions, elementary equivalence and isomorphism and partial isomorphisms and bounded quantifier rank.
Read as: Gamma union the set of all inequalities c sub i not equal to c sub j for distinct i and j
Means: The expression reads: Gamma union the set of all inequalities c sub i not equal to c sub j for distinct i and j. It names the displayed model-theoretic object in the surrounding argument. Its 3 exact source occurrences were reviewed in the compactness argument for an infinite model.
Read as: q is in I
Means: The expression reads: q is in I. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 2 exact source occurrences were reviewed in partial isomorphisms and bounded quantifier rank.
Read as: n equals two times r
Means: The expression reads: n equals two times r. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: structure Q
Means: The expression reads: structure Q. It names the displayed model-theoretic object in the surrounding argument. Its 1 exact source occurrence was reviewed in dense linear orders without endpoints.
Read as: X equals the set of h open parenthesis x close parenthesis such that x is in X
Means: The expression reads: X equals the set of h open parenthesis x close parenthesis such that x is in X. It identifies a theory, a set defined by a condition, or membership in such a set. Its 1 exact source occurrence was reviewed in elementary equivalence and isomorphism.
Read as: relation I sub n open parenthesis sequence a comma sequence b close parenthesis
Means: The expression reads: relation I sub n open parenthesis sequence a comma sequence b close parenthesis. It belongs to the finite back-and-forth analysis that connects sequence relations with formulas of bounded quantifier rank. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: h open parenthesis the interpretation of c in structure M close parenthesis equals the interpretation of c in structure M prime
Means: The expression reads: h open parenthesis the interpretation of c in structure M close parenthesis equals the interpretation of c in structure M prime. It identifies an interpretation or term value in a named structure and assignment. Its 1 exact source occurrence was reviewed in elementary equivalence and isomorphism.
Read as: relation I sub n plus one open parenthesis sequence a comma sequence b close parenthesis
Means: The expression reads: relation I sub n plus one open parenthesis sequence a comma sequence b close parenthesis. It belongs to the finite back-and-forth analysis that connects sequence relations with formulas of bounded quantifier rank. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: n is greater than or equal to m
Means: The expression reads: n is greater than or equal to m. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in the compactness argument for an infinite model.
Read as: p is in I
Means: The expression reads: p is in I. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 2 exact source occurrences were reviewed in partial isomorphisms and bounded quantifier rank.
Read as: structure N satisfies formula A
Means: The expression reads: structure N satisfies formula A. It asserts truth or failure of truth in the named structure, with the displayed assignment when one is present. Its 1 exact source occurrence was reviewed in the theory of a structure.
Read as: h is a function from the domain of structure M to the domain of structure M prime
Means: The expression reads: h is a function from the domain of structure M to the domain of structure M prime. It declares the displayed function together with its domain and codomain. Its 1 exact source occurrence was reviewed in elementary equivalence and isomorphism.
Read as: structure M prime satisfies formula A at assignment h composed with s
Means: The expression reads: structure M prime satisfies formula A at assignment h composed with s. It asserts truth or failure of truth in the named structure, with the displayed assignment when one is present. Its 1 exact source occurrence was reviewed in elementary equivalence and isomorphism.
Read as: the domain of structure M equals the set containing a sub zero comma a sub one comma and so on
Means: The expression reads: the domain of structure M equals the set containing a sub zero comma a sub one comma and so on. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: the order relation less than sub one equals the interpretation of the order relation less than in structure M sub one
Means: The expression reads: the order relation less than sub one equals the interpretation of the order relation less than in structure M sub one. It identifies an interpretation or term value in a named structure and assignment. Its 1 exact source occurrence was reviewed in dense linear orders without endpoints.
Read as: p sub n plus one is in I
Means: The expression reads: p sub n plus one is in I. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 2 exact source occurrences were reviewed in partial isomorphisms and bounded quantifier rank.
Read as: structure R is elementarily equivalent to structure S
Means: The expression reads: structure R is elementarily equivalent to structure S. It compares structures by isomorphism, partial isomorphism, or agreement on first-order sentences, exactly as the notation indicates. Its 1 exact source occurrence was reviewed in dense linear orders without endpoints.
Read as: not formula A
Means: The expression reads: not formula A. It is a complete first-order condition; quantifier scope and connective order are spoken explicitly. Its 1 exact source occurrence was reviewed in the compactness argument for an infinite model.
Read as: relation I sub n holds of the sequence formed by appending a to sequence a and the sequence formed by appending b to sequence b
Means: The expression reads: relation I sub n holds of the sequence formed by appending a to sequence a and the sequence formed by appending b to sequence b. It belongs to the finite back-and-forth analysis that connects sequence relations with formulas of bounded quantifier rank. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: the empty set belongs to the family I of partial isomorphisms
Means: The expression reads: the empty set belongs to the family I of partial isomorphisms. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in dense linear orders without endpoints.
Read as: a
Means: The expression reads: a. It names the displayed model-theoretic object in the surrounding argument. Its 2 exact source occurrences were reviewed in partial isomorphisms and bounded quantifier rank.
Read as: for every x, for every y, for every z, open parenthesis x is less than y implies open parenthesis y is less than z implies x is less than z close parenthesis close parenthesis
Means: The expression reads: for every x, for every y, for every z, open parenthesis x is less than y implies open parenthesis y is less than z implies x is less than z close parenthesis close parenthesis. It is a complete first-order condition; quantifier scope and connective order are spoken explicitly. Its 1 exact source occurrence was reviewed in dense linear orders without endpoints.
Read as: the interpretation of c in structure M equals the interpretation of c in structure M prime
Means: The expression reads: the interpretation of c in structure M equals the interpretation of c in structure M prime. It identifies an interpretation or term value in a named structure and assignment. Its 2 exact source occurrences were reviewed in reducts and expansions and substructures.
Read as: h open parenthesis the value of t in structure M at assignment s close parenthesis equals h open parenthesis the interpretation of c in structure M close parenthesis equals the interpretation of c in structure M prime
Means: The expression reads: h open parenthesis the value of t in structure M at assignment s close parenthesis equals h open parenthesis the interpretation of c in structure M close parenthesis equals the interpretation of c in structure M prime. It identifies an interpretation or term value in a named structure and assignment. Its 1 exact source occurrence was reviewed in elementary equivalence and isomorphism.
Read as: structure R is elementarily equivalent to structure Q
Means: The expression reads: structure R is elementarily equivalent to structure Q. It compares structures by isomorphism, partial isomorphism, or agreement on first-order sentences, exactly as the notation indicates. Its 1 exact source occurrence was reviewed in dense linear orders without endpoints.
Read as: structure M prime
Means: The expression reads: structure M prime. It names the displayed model-theoretic object in the surrounding argument. Its 7 exact source occurrences were reviewed in reducts and expansions, substructures and elementary equivalence and isomorphism.
Read as: t is syntactically identical to f open parenthesis t sub one comma and so on comma t sub n close parenthesis
Means: The expression reads: t is syntactically identical to f open parenthesis t sub one comma and so on comma t sub n close parenthesis. It states which constant, variable, or compound term the term t is syntactically identical to. Its 1 exact source occurrence was reviewed in elementary equivalence and isomorphism.
Read as: the quantifier rank of formula A is in the natural numbers
Means: The expression reads: the quantifier rank of formula A is in the natural numbers. It belongs to the finite back-and-forth analysis that connects sequence relations with formulas of bounded quantifier rank. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: open parenthesis h composed with s close parenthesis open parenthesis x close parenthesis equals h open parenthesis s open parenthesis x close parenthesis close parenthesis
Means: The expression reads: open parenthesis h composed with s close parenthesis open parenthesis x close parenthesis equals h open parenthesis s open parenthesis x close parenthesis close parenthesis. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in elementary equivalence and isomorphism.
Read as: b sub i is less than sub two b is less than sub two b sub i plus one
Means: The expression reads: b sub i is less than sub two b is less than sub two b sub i plus one. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in dense linear orders without endpoints.
Read as: structure M is isomorphic to structure M prime
Means: The expression reads: structure M is isomorphic to structure M prime. It compares structures by isomorphism, partial isomorphism, or agreement on first-order sentences, exactly as the notation indicates. Its 1 exact source occurrence was reviewed in elementary equivalence and isomorphism.
Read as: structure R is elementarily equivalent to structure S
Means: The expression reads: structure R is elementarily equivalent to structure S. It compares structures by isomorphism, partial isomorphism, or agreement on first-order sentences, exactly as the notation indicates. Its 1 exact source occurrence was reviewed in the theory of a structure.
Read as: a sub i is less than sub one a is less than sub one a sub i plus one
Means: The expression reads: a sub i is less than sub one a is less than sub one a sub i plus one. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in dense linear orders without endpoints.
Read as: the set of all finite sequences over the domain of structure M
Means: The expression reads: the set of all finite sequences over the domain of structure M. It denotes a structure domain, a function domain, or the finite sequences drawn from that domain. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: x sub one
Means: The expression reads: x sub one. It names the displayed model-theoretic object in the surrounding argument. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: structure M is isomorphic to structure N
Means: The expression reads: structure M is isomorphic to structure N. It compares structures by isomorphism, partial isomorphism, or agreement on first-order sentences, exactly as the notation indicates. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: is less than
Means: The expression reads: is less than. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 2 exact source occurrences were reviewed in the theory of a structure and dense linear orders without endpoints.
Read as: the theory of structure M
Means: The expression reads: the theory of structure M. It identifies a theory, a set defined by a condition, or membership in such a set. Its 3 exact source occurrences were reviewed in the theory of a structure.
Read as: structure M satisfies formula A
Means: The expression reads: structure M satisfies formula A. It asserts truth or failure of truth in the named structure, with the displayed assignment when one is present. Its 3 exact source occurrences were reviewed in elementary equivalence and isomorphism and the theory of a structure.
Read as: i
Means: The expression reads: i. It names the displayed model-theoretic object in the surrounding argument. Its 2 exact source occurrences were reviewed in elementary equivalence and isomorphism and dense linear orders without endpoints.
Read as: structure M satisfies the negation of formula A
Means: The expression reads: structure M satisfies the negation of formula A. It asserts truth or failure of truth in the named structure, with the displayed assignment when one is present. Its 1 exact source occurrence was reviewed in the theory of a structure.
Read as: language L prime equals the union of language L and the singleton set containing P
Means: The expression reads: language L prime equals the union of language L and the singleton set containing P. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in reducts and expansions.
Read as: the interpretation of R in structure N equals the interpretation of R in structure M intersect N superscript n
Means: The expression reads: the interpretation of R in structure N equals the interpretation of R in structure M intersect N superscript n. It identifies an interpretation or term value in a named structure and assignment. Its 1 exact source occurrence was reviewed in substructures.
Read as: h open parenthesis the interpretation of f in structure M open parenthesis a sub one comma and so on comma a sub n close parenthesis close parenthesis equals the interpretation of f in structure M prime open parenthesis h open parenthesis a sub one close parenthesis comma and so on comma h open parenthesis a sub n close parenthesis close parenthesis
Means: The expression reads: h open parenthesis the interpretation of f in structure M open parenthesis a sub one comma and so on comma a sub n close parenthesis close parenthesis equals the interpretation of f in structure M prime open parenthesis h open parenthesis a sub one close parenthesis comma and so on comma h open parenthesis a sub n close parenthesis close parenthesis. It identifies an interpretation or term value in a named structure and assignment. Its 1 exact source occurrence was reviewed in elementary equivalence and isomorphism.
Read as: t
Means: The expression reads: t. It names the displayed model-theoretic object in the surrounding argument. Its 2 exact source occurrences were reviewed in elementary equivalence and isomorphism.
Read as: y is in the domain of structure M prime
Means: The expression reads: y is in the domain of structure M prime. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in elementary equivalence and isomorphism.
Read as: the interpretation of P in structure M prime equals R
Means: The expression reads: the interpretation of P in structure M prime equals R. It identifies an interpretation or term value in a named structure and assignment. Its 1 exact source occurrence was reviewed in reducts and expansions.
Read as: for every x, there exists y, y is less than x
Means: The expression reads: for every x, there exists y, y is less than x. It is a complete first-order condition; quantifier scope and connective order are spoken explicitly. Its 1 exact source occurrence was reviewed in dense linear orders without endpoints.
Read as: P is in language L
Means: The expression reads: P is in language L. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in reducts and expansions.
Read as: language L
Means: The expression reads: language L. It names the displayed model-theoretic object in the surrounding argument. Its 4 exact source occurrences were reviewed in reducts and expansions, substructures and elementary equivalence and isomorphism.
Read as: the domain of structure N equals the set containing b sub zero comma b sub one comma and so on
Means: The expression reads: the domain of structure N equals the set containing b sub zero comma b sub one comma and so on. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: the theory of structure M is a subset of the theory of structure N
Means: The expression reads: the theory of structure M is a subset of the theory of structure N. It identifies a theory, a set defined by a condition, or membership in such a set. Its 1 exact source occurrence was reviewed in the theory of a structure.
Read as: left angle bracket a sub one comma and so on comma a sub n right angle bracket is in the interpretation of P in structure M
Means: The expression reads: left angle bracket a sub one comma and so on comma a sub n right angle bracket is in the interpretation of P in structure M. It identifies an interpretation or term value in a named structure and assignment. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: the interpretation of P in structure M equals the interpretation of P in structure M prime
Means: The expression reads: the interpretation of P in structure M equals the interpretation of P in structure M prime. It identifies an interpretation or term value in a named structure and assignment. Its 1 exact source occurrence was reviewed in reducts and expansions.
Read as: the domain of structure N equals N
Means: The expression reads: the domain of structure N equals N. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in substructures.
Read as: the negation of formula A belongs to the theory of structure M
Means: The expression reads: the negation of formula A belongs to the theory of structure M. It identifies a theory, a set defined by a condition, or membership in such a set. Its 1 exact source occurrence was reviewed in the theory of a structure.
Read as: language L prime
Means: The expression reads: language L prime. It names the displayed model-theoretic object in the surrounding argument. Its 2 exact source occurrences were reviewed in reducts and expansions.
Read as: structure M satisfies formula A at assignment sequence a
Means: The expression reads: structure M satisfies formula A at assignment sequence a. It asserts truth or failure of truth in the named structure, with the displayed assignment when one is present. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
Read as: the expansion of structure M by relation R
Means: The expression reads: the expansion of structure M by relation R. It names the displayed model-theoretic object in the surrounding argument. Its 1 exact source occurrence was reviewed in reducts and expansions.
Read as: structure M is partially isomorphic to structure N
Means: The expression reads: structure M is partially isomorphic to structure N. It compares structures by isomorphism, partial isomorphism, or agreement on first-order sentences, exactly as the notation indicates. Its 3 exact source occurrences were reviewed in partial isomorphisms and bounded quantifier rank.
Read as: the interpretation of f in structure M equals the interpretation of f in structure M prime
Means: The expression reads: the interpretation of f in structure M equals the interpretation of f in structure M prime. It identifies an interpretation or term value in a named structure and assignment. Its 1 exact source occurrence was reviewed in reducts and expansions.
Read as: finite sequence b followed by element b
Means: The expression reads: finite sequence b followed by element b. It names the displayed model-theoretic object in the surrounding argument. Its 4 exact source occurrences were reviewed in partial isomorphisms and bounded quantifier rank.
Read as: a is in the domain of structure M
Means: The expression reads: a is in the domain of structure M. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 2 exact source occurrences were reviewed in partial isomorphisms and bounded quantifier rank.
Read as: s sub two open parenthesis x sub i close parenthesis equals b sub i
Means: The expression reads: s sub two open parenthesis x sub i close parenthesis equals b sub i. It states the displayed equality, inclusion, membership, or order relation between the named objects. Its 1 exact source occurrence was reviewed in partial isomorphisms and bounded quantifier rank.
A smaller-language structure is the reduct of a larger-language structure precisely when their domains and all shared symbol interpretations agree.
A structure and its expansion satisfy exactly the same sentences from the smaller language.
Unsolved exercise asking for a proof that reducts preserve all sentences of the smaller language.
Defines the notation for expanding a structure by interpreting one newly added predicate as a chosen relation.
A substructure has a smaller domain, agrees on constants, restricts functions, and agrees on predicates over its domain.
Arbitrarily large finite models imply the existence of an infinite model.
No single first-order sentence is true exactly in structures with infinite domain.
Two structures in the same language are elementarily equivalent when they agree on every sentence.
An isomorphism is a bijection of domains preserving constants, predicates, and functions.
Isomorphism preserves every first-order sentence.
Two aligned equations compare a compound term in M and M prime, with a disclosed wrong-structure correction in the second row.
A four-line equality chain proves that an isomorphism carries compound-term values to their values in the target structure, with a disclosed parenthesis correction.
Unsolved exercise asking for the formula case of isomorphism invariance and an accounting of all five isomorphism conditions.
An automorphism is an isomorphism from a structure onto itself.
Unsolved exercise asking to prove that every definable subset is fixed setwise by every automorphism.
The theory of M is the set of all sentences true in M.
Every structure decides each sentence, so its theory is complete.
Any structure satisfying every sentence true in M is elementarily equivalent to M.
A finite partial function is a partial isomorphism when it satisfies injectivity and the defined constant, predicate, and function preservation clauses.
Two structures are partially isomorphic when a nonempty family of partial isomorphisms has the forth and back extension properties.
For enumerable structures, a back-and-forth family can be assembled into a total isomorphism.
Unsolved exercise asking to verify that the map constructed in the preceding theorem is an isomorphism.
For purely relational languages, partially isomorphic structures agree on every first-order sentence.
Quantifier rank measures nesting depth, and n equivalent structures agree on every sentence up to that rank.
A finite relational language has only finitely many sentences of any fixed maximum quantifier rank, up to logical equivalence.
Defines notation for finite sequences and for appending one element to a sequence.
Defines relations I sub n first by atomic agreement and then by mutual one-element extensions.
M and N are approximately equivalent at level n when I sub n relates their empty sequences.
I sub n implies agreement on formulas of rank at most n, with a converse for finite languages.
In finite relational languages, level-n back-and-forth equivalence is exactly agreement through quantifier rank n.
Six first-order axioms specify irreflexivity, transitivity, totality, no endpoints, and density.
Any two enumerable dense linear orders without endpoints are isomorphic.
Unsolved exercise asking to finish the Cantor back-and-forth proof by establishing the Back clause.